Fixed points theorems for unsaturated and saturated classes of contractive mappings in Banach spaces

TL;DR

This paper introduces the concepts of saturated and unsaturated classes of contractive mappings in Banach spaces, showing how enriching techniques yield new fixed point results for unsaturated classes and identifying classes that cannot be enlarged.

Contribution

It defines saturated classes of contractive mappings and demonstrates how enriching techniques produce new fixed point theorems for unsaturated classes, advancing metric fixed point theory.

Findings

Enriching contractive mappings yields new fixed point results for unsaturated classes.

Saturated classes cannot be enlarged by enriching techniques.

Survey of recent fixed point results for five unsaturated classes.

Abstract

Based on the technique of enriching contractive type mappings, a technique that has been used successfully in some recent papers, we introduce the concept of {\it saturated} class of contractive mappings. We show that, from this perspective, the contractive type mappings in the metric fixed point theory can be separated into two distinct classes, unsaturated and saturated, and that, for any unsaturated class of mappings, the technique of enriching contractive type mappings provides genuine new fixed point results. We illustrate the concept by surveying some significant fixed point results obtained recently for five remarkable unsaturated classes of contractive mappings. In the second part of the paper, we also identify some classes of saturated classes of contractive mappings, whose main feature is that they cannot be enlarged by means of the technique of enriching the contractive mappings.